Analytical study of Numerical Approximation for Ordinary Differential Equation

Abstract

This research primarily focuses on methods for solving Initial Value Problems (IVPs) for Ordinary Differential Equations (ODEs), with a particular emphasis on three key methods: the Euler method, the Fourth-Order Runge-Kutta method (RK4), and the Fifth-Order Runge-Kutta method (RK5). The latter two methods are highlighted as highly effective and practical for solving such problems. To ensure accuracy, the numerical solutions obtained using these methods were compared with exact solutions. The results showed that the numerical solutions agreed well with the exact solutions. Additionally, numerical comparisons between the Euler method and the Runge-Kutta methods were demonstrated using MATLAB. Finally, the errors of the proposed methods were studied and calculated for various examples.